Quantum Fourier Transform
Introduction
The Quantum Fourier Transform (QFT) is a linear transformation on quantum bits and is a part of many quantum algorithms, notably Shor's algorithm and the quantum phase estimation algorithm. It is the quantum analogue to the classical Fourier transform. The quantum Fourier transform is a key tool in the design of quantum algorithms and can be used to analyze quantum states and operations.
Definition
The Quantum Fourier Transform is defined analogously to the discrete Fourier transform, but with a change of basis from the computational basis to the Fourier basis. The Fourier basis states are superpositions of the computational basis states with different phases. The QFT maps the basis state |j⟩ to a superposition of all basis states, with the phase of each state |k⟩ proportional to jk.
Mathematical Representation
The mathematical representation of the Quantum Fourier Transform is as follows:
QFT: |j⟩ ⟼ 1/√N ∑_{k=0}^{N-1} e^{2πijk/N} |k⟩
where N is the total number of states, j and k are the basis states, and e is the base of the natural logarithm.
Quantum Circuit Representation
The Quantum Fourier Transform can be implemented in a quantum circuit using a combination of Hadamard gates and controlled phase shift gates. The circuit begins with a series of Hadamard gates applied to each qubit, followed by a series of controlled phase shift gates. The controlled phase shift gates introduce the necessary phases to each basis state. The final step of the circuit is a series of swap gates to reverse the order of the qubits.
Applications
The Quantum Fourier Transform is used in many quantum algorithms, including:
- Shor's algorithm: The QFT is used in Shor's algorithm to find the period of a function, which is a crucial step in factoring large numbers.
- Quantum phase estimation algorithm: The QFT is used in the quantum phase estimation algorithm to estimate the phase of an eigenstate of a unitary operator.
- Quantum amplitude estimation: The QFT is used in quantum amplitude estimation to estimate the amplitude of a state in a quantum register.
See Also
- Quantum algorithm - Quantum computing - Quantum information theory